Central Kumamoto is best covered by the city's two tram lines — slow but very cheap, and they connect most of the sights you will want to visit (Kumamoto Castle, the shopping arcades, Suizenji Garden, Kumamoto Station). A one-day tram pass costs around 500 yen and pays for itself on the first three rides.
The city has an extensive bus network for places off the tram routes (Honmyō-ji, Tatsuda) and a free shuttle around the castle perimeter. Bicycles can be rented at Kumamoto Station and at several locations in the centre. Taxis are plentiful and not expensive by Tokyo standards.